表面多次波最小二乘逆时偏移成像

刘学建， 刘伊克

1 中国科学院地质与地球物理研究所工程地质力学重点实验室， 北京 100029 2 中国科学院大学， 北京 100049



表面多次波最小二乘逆时偏移成像

刘学建1,2， 刘伊克1

1 中国科学院地质与地球物理研究所工程地质力学重点实验室， 北京 100029 2 中国科学院大学， 北京 100049

使用相同的炮记录，多次波偏移能提供比反射波偏移更广的地下照明和更多的地下覆盖但是同时产生很多的串声噪声.相比传统逆时偏移，最小二乘逆时偏移反演的反射波成像结果具有更高的分辨率和更均衡的振幅.我们主要利用最小二乘逆时偏移压制多次波偏移产生的串声噪声.多次波最小二乘逆时偏移通常需要一定的迭代次数以较好地消除串声噪声.若提前将一阶多次波从所有阶数的多次波中过滤出来，使用相同的迭代次数，一阶多次波的最小二乘逆时偏移能够得到具有更高信噪比的成像剖面，而且能够提供与多次波最小二乘逆时偏移相似的有效地下结构成像.

最小二乘逆时偏移； 多次波成像； 一阶多次波

In order to invert primaries as an image, LSRTM iteratively solves a misfit function that is the L2 norm of the amplitude residual between the modeled and observed primaries. Born modeling is a linear two-step procedure and synthesizes primaries perturbed by an image, which bases the LSRTM. Conventional RTM is the adjoint of Born modeling, whereas the analytical solution of the misfit function is the generalized-inverse of the Born modeling. The analytical solution is hard to be obtained because the Hessian matrix is so large, so a nonlinear optimal scheme, e.g., the steepest-descent method, can be used to iteratively solve the misfit function. Taking the released Sigsbee2b data as an example, we can intuitively conclude that LSRTM provides an image with higher resolution and more balanced amplitude and suppresses the migration artifacts compared with conventional RTM. Different with the misfit function for the conventional LSRTM, the misfit function for the LSRTM of multiples is the L2 norm of the amplitude residual between the modeled multiples and estimated multiples during the regular seismic data processing. The accurate calculation for the modeling of multiples is crucial for the success of this method, where a modified Born modeling procedure and an accurate background velocity are utilized. Instead of a point source, the recorded data including primaries and multiples are forward propagated and stacked as the downgoing wavefield. Each discrete point of the image is seen as a scatter. The two-order time derivative of downgoing wavefield is scattered by the RTM image of multiples, and upgoing wavefield is the stack of scattered waves. Surface-related multiples are modeled by recording the upgoing wavefield at receivers. Similar to the conventional LSRTM, LSRTM of multiples also can iteratively seek the reflectivity model using a nonlinear optimal method. Moreover, to invert first-order multiples as an image, the misfit function based on the L2 norm of the amplitude residual between the observed and Born modeled first-order multiples should be built. Compared with the Born modeling of all-order multiples, instead of total recorded data, only primaries are forward propagated for the Born modeling of first-order multiples. The observed first-order multiples are estimated by a modified SRME, which includes two steps: (1) predicting higher-order multiples by the convolution of primaries and multiples; (2) adaptively subtracting higher-order multiples from all multiples.

RTM of all-order multiples, LSRTM of all-order multiples, RTM of first-order multiples and LSRTM of first-order multiples have been tested on a three-layer and the Marmousi2 model. Only 16 shot gathers are used for imaging on the three-layer model. RTM image of multiples provide wider illumination and higher fold for subsurface, whereas there are a lot of artifacts in the image of multiples. After 10 iterations, LSRTM attenuate most of the artifacts in the image of multiples, except the artifacts at bottom. Moreover, LSRTM of first-order multiples provide a more cleaner section than LSRTM of all-order multiples. There are artifacts in the modeled data using the RTM image of multiples, whereas the modeled data using LSRTM image of multiples have a good match with the estimated multiples using SRME and avoid most artifacts. On the Marmousi2 model, there are many artifacts in the RTM image of multiples, which are mostly attenuated by LSRTM after 5 iterations. However, after 5 iterations, there are still residual artifacts in the LSRTM image of multiples, which disappear in LSRTM image of first-order multiples. On above two experiments, LSRTM of multiples and LSRTM of first-order multiples both converge very fast and robust.

Compared with RTM, LSRTM provides image with more balanced amplitude and better resolution and suppresses the migration artifacts. RTM of multiples can provide a wider illumination and higher fold for subsurface. However, there are many crosstalk artifacts in the RTM image of all-order multiples. LSRTM can attenuate most of the crosstalk artifacts in the image of multiples but costs huge computation of many iterations. A modified SRME is proposed to filter first-order multiples. With the same iterations used, LSRTM of first-order multiples provide a much cleaner section, and provider a similar true image of reflectors compared with LSRTM of all-order multiples. Prior to LSRTM of first-order multiples, first-order multiples are needed to be estimated by a modified SRME.

1 引言

多次波通常被认为是一种噪声，并且在偏移之前的数据预处理中尽可能的减掉 (Berkhout and Verschuur, 1997; Verschuur and Berkhout, 1997; Liu et al., 2009, 2010; Dragoset et al., 2010；李鹏等，2007；王维红和井洪亮，2015).实际上多次波在地下比反射波传播路径更长且覆盖范围更广，多次波中含有丰富的小角度信息.在使用相同炮记录偏移时，多次波能为下地表提供更宽的成像范围和更多的覆盖.近年来，很多的学者致力于多次波成像的研究，并且提出了多种多次波成像方法.多次波可以首先被转化为反射波(Berkhout and Verschuur, 2003, 2006; Schuster et al., 2004; Verschuur and Berkhout, 2005； 刘学建等，2015)，并利用传统的逆时偏移方法成像.更一步的，反射波波动方程偏移方法或者逆时偏移成像方法可以修改为多次波直接波动方程(Guitton, 2002; Muijs et al., 2007; Lu et al., 2011)或者逆时偏移(Liu et al., 2011a, 2011b)成像方法.逆时偏移(Baysal et al., 1983)是一种强有力的成像技术，能够利用多种地震波 (包括反射波、回转波以及棱柱波)，从而对速度的横向变化有良好的适应性并有能力对陡倾角成像.多次波逆时偏移也具有上述传统逆时偏移成像的优势.然而，因为不同阶数多次波波场之间的互相关，多次波逆时偏移成像过程中将会产生大量的串声噪声.这些串声噪声分布在整个成像剖面中，破坏了有效成像的结构和振幅.串声噪声很难消除并且大大降低了多次波成像的价值.

相对于传统的偏移方法，最小二乘逆时偏移(Dong et al., 2012; Dai et al., 2012; Dai and Schuster, 2013; Zhang et al., 2015)能提供振幅更均衡、分辨率更高的反射波成像结果，并能够消除偏移噪声.最小二乘逆时偏移方法也能够消除多次波成像中的串声噪声(Brown and Guitton, 2005; Wong et al., 2014; Zhang and Schuster, 2014)，其目标函数为波恩模拟的多次波与观测的多次波之间的差的L2 范数.通过一个最优化迭代算法(如最速下降法)求解该目标函数以得到地下反射率分布的过程，即为多次波的最小二乘逆时偏移反演成像.多次波与反射波的波恩模拟区别主要在于：不是利用震源子波，而是将包含反射波和多次波的观测数据作为震源正传.最小二乘逆时偏移每次迭代都消耗大约几倍逆时偏移的计算量，计算成本非常高.而多次波最小二乘逆时偏移往往需要一定数量的迭代次数以较好地压制串声噪声.因此，我们修改SRME方法，只将一阶多次波从所有阶数的多次波中过滤出来.基于波恩模拟的一阶多次波与记录的一阶多次波差的二范数最小，一阶多次波的最小二乘逆时偏移成像方法能够以相同的迭代次数得到更高信噪比的多次波成像剖面.

本文首先回顾了反射波最小二乘逆时偏移的基本原理，并通过Sigsbee2b模型来验证其优势.然后阐述了多次波最小二乘逆时偏移的基本原理；一阶多次波的分离方案；一阶多次波的最小二乘逆时偏移原理.最后利用一个三层模型及Marmousi2模型，对多次波及一阶多次波最小二乘逆时偏移进行数值实验.

2 基本原理

2.1 反射波最小二乘逆时偏移

对于二维模型，检波器记录到的从震源激发的地下一次散射波，可以通过波恩近似来模拟，其频率域的表达式为：

d(xr,xs,ω)= ω2∫G0(xr,x,ω)r(x)G0(x,xs,ω)

×fs(ω)dx.

(1)

其中，ω表示圆频率，fs(ω)表示震源子波，G0(xr,x,ω)和G0(x,xs,ω)分别表示连接检波器xr和震源xs与地下散射点x=(x,z)的格林函数，r(x)表示反射率分布模型，d(xr,xs,ω)为模拟的散射波.波恩模拟的向量表达式为：

(2)

而传统的偏移方法可以认为是波恩模拟的共轭转置：

(3)

(4)

用算子M(r(x))表示时间域的波恩模拟，其具体实现方法为：

(5)

其中，v0(x)为光滑的背景速度，p0(x,t)为下行的震源波场，pr(x,t)为上行的波场.用算子MT(dobs)表示时间域的逆时偏移，其实现过程简单概括为：

(6)

(7)

(8)

其中，q(x,t)为检波器数据的逆传波场.另外，为满足成像条件(8)的要求，公式(6)模拟的震源波场需要被重建为时间逆序的波场.

反射波最小二乘逆时偏移最为基本的目标函数为波恩模拟的反射波d(xr,xs,t)与观测的反射波dobs(xr,xs,t)之间差的能量：

f(r(x))=∫∫(d(xr,xs,t)-dobs(xr,xs,t))2dtdxr,

(9)

第k次迭代模拟的反射波d(k)(xr,xs,t)和数据残差δd(k)(xr,xs,t)表示为：

(10)

δd(k)(xr,xs,t)=d(k)(xr,xs,t)-dobs(xr,xs,t).

(11)

目标函数(9)的梯度和基于梯度下降法的迭代解分别为：

(12)

(13)

如公式(10)—(13)所示，最小二乘逆时偏移是一个迭代求解过程.如图1的对比(使用Sigsbee2b发布的层速度和带有鬼波的反射波数据)，相比传统逆时偏移，最小二乘逆时偏移成像结果具有较高分辨率、更均衡的振幅，并能压制偏移噪声.

2.2 多次波最小二乘逆时偏移

相对于子波震源，采集的包含反射波dobs(xr,xs,t)和多次波mobs(xr,xs,t)的全波波场记录Dobs(xr,xs,t)可以看作多次波的二次震源，则波恩模拟的多次波m(xr,xs,t)可以表示为

(14)

相应的，多次波的逆时偏移可以简单表示为：

(16)

(17)

地震数据处理流程中，反射波与多次波将会被分离，分离出的多次波作为观测的多次波.多次波的最小二乘逆时偏移的目标函数为波恩模拟的多次波与观测的多次波之间差的能量：

图1 将SMAART JV发布的带有鬼波的反射波数据作为观测反射波，且将层速度平滑后作为背景速度(a)反射波逆时偏移成像结果；(b)反射波最小二乘逆时偏移(30次迭代)成像结果. 图(b)具有更高的分辨率，如黑色箭头所示，散射体有更好的聚焦.图(b)在盐丘下有更均衡的振幅.如白色箭头所示，最小二乘逆时偏移能够压制偏移噪声.Fig.1 The released primaries with ghosts by SMAART JV are treated by observed primaries, and the interval velocity is smoothed as to a background velocity(a) RTM image of primaries; (b) LSRTM of primaries with 30 iterations. Figure (b) have a better resolution, as indicated by black arrows, the scatters are better focused. Figure (b) have a more balance amplitude at the subsurface. As indicated by white arrows, LSRTM can suppress migration noises.

f(r(x))=∫∫(m(xr,xs,t)-mobs(xr,xs,t))2dtdxr.

(18)

如公式(10)—(13)所示, 一个相似的迭代过程求解目标函数，则得到多次波的最小二乘逆时偏移反演成像结果.

2.3 一阶多次波最小二乘逆时偏移

采集的反射波数据dobs(xr,xs,t)可以看作一阶多次波的二次震源，则波恩模拟的一阶多次波m1(xr,xs,t)可以表示为

(19)

(21)

(22)

(23)如公式(10)—(13)所示, 一个相似的迭代过程求解目标函数，则得到一阶多次波的最小二乘逆时偏移反演成像结果.

3 数值实验

如图2，本次实验的流程是首先正演带多次波的数据，用SRME分离反射波(含鬼波和层间多次波)和表面多次波，一个修改的SRME流程从所有多次波中分离出一阶多次波.反射波，表面多次波和分离出的一阶多次波可以应用于最小二乘逆时偏移反演成像中.

3.1 简单三层模型

如图3所示为一个简单三层声波速度模型，横向1201网格点，纵向501网格点，网格间距5 m.共有16炮用于偏移成像;震源子波主频为15 Hz,并等间距的在2.04 km和3.84 km之间激发.中间放炮观测系统，每个炮记录有201个检波器.震源和检波器的深度为5 m.最大记录时间长度和采样间隔分别为3 s和2 ms.

图2 实验流程Fig.2 The workflow of the experiments

图3 三层声波速度模型Fig.3 Three-layer acoustic velocity model

图4 (a) SRME估计的反射波的逆时偏移成像结果；(b)多次波逆时偏移成像结果多次波偏移为下地表提供了更宽的照明范围和更多的覆盖次数；如箭头所示，多次波偏移也产生了很多的串声假象.Fig.4 (a) RTM image of primaries estimated by SRME; (b) RTM image of multiplesMigration of multiples provides wider illumination and more fold for subsurface; however, as indicated by the arrows, migration of multiples also generates many crosstalk artifacts.

图5 (a)多次波最小二乘逆时偏移成像结果(10次迭代)； (b)一阶多次波最小二乘逆时偏移成像结果(10次迭代)如(a)中蓝色箭头所示，多次波最小二乘逆时偏移压制了图4b中大部分的串声假象.如黑色箭头所示，(a)中残留在深部的串声假象在(b)中消失.而且(b)与(a)有相似的能对应地下反射位置的有效成像结果.Fig.5 (a) LSRTM image of all-order multiples (10 iterations); (b) LSRTM image of first-order multiples (10 iterations)As indicated by blue arrows in (a), LSRTM of multiples suppresses most of crosstalk artifacts in Fig.4b. As indicated by black arrows, residual artifacts at deep in (a) disappear in (b). Moreover, panel (b) provides a similar true-image of reflectors to panel (a).

如图4所示为反射波成像与多次波成像的对比.显而易见，当相同的炮记录用于偏移时，多次波偏移为下地表提供了更宽的照明范围和更多的覆盖次数；然而，多次波偏移也产生了很多的串声假象.如图5为多次波最小二乘逆时偏移与一阶多次波最小二乘逆时偏移的对比，它们都用了10次迭代计算.多次波最小二乘逆时偏移压制了大部分多次波逆时偏移中的串声假象；然而，在多次波最小二乘偏移剖面的深部，仍有残留的串声假象；这些残留的串声假象在一阶多次波最小二乘逆时偏移剖面中消失.使用同样的迭代次数，一阶多次波最小二乘逆时偏移能够提供与多次波最小二乘逆时偏移相似的地下构造成像结果；而一阶多次波最小二乘逆时偏移结果有更高的信噪比.

另外，我们通过数据域的对比来说明，多次波最小二乘逆时偏移能够消除多次波逆时偏移中的串声假象.如图6 所示，利用多次波逆时偏移结果，波恩模拟的多次波中有很多的虚假同相轴；而多次波最小二乘逆时偏移结果，波恩模拟的多次波没有虚假的同相轴，与SRME估计的多次波有很好的匹配.图7中为多次波及一阶多次波最小二乘逆时偏移中归一化的数据残差收敛曲线，它们表现出相似的快速稳定收敛性质.

图6 (a) SRME估计的所有阶数的多次波；(b)波恩模拟的多次波利用如图4b所示的多次波偏移结果；(c)波恩模拟的多次波利用如图5a所示的多次波最小二乘逆时偏移结果如箭头所示，(b)中虚假的同相轴在(c)中消失.(c)中模拟的多次波与(a)中多次波有较好的匹配.Fig.6 (a) Estimated all-order multiples using SRME; (b) Born modeled multiples using the RTM image of all-order multiples in Fig.4b; (c) Born modeled multiples using the LSRTM image of all-order multiples in Fig.5aAs indicated by the arrows, the false events in (b) disappear in (c). The modeled multiples in (c) have a good match with multiples in (a).

图7 简单三层模型上多次波(实线)及一阶多次波(散点)最小二乘逆时偏移的归一化数据残差收敛曲线Fig.7 Normalized data residual for LSRTM of multiples (solid line) and first-order multiples (dots) on the simple three-layer model

3.2 Marmousi2模型

如图8所示为Marmousi2声波模型的中间部分，横向1601网格点，纵向561网格点，网格间距6.25 m.共有81炮用于偏移成像;震源子波主频为20 Hz,并等间距的在2 km和8 km之间激发.中间放炮观测系统，每个炮记录有241个检波器.震源和检波器的深度为6.25 m.最大记录时间长度和采样间隔分别为4 s和2 ms.

图9对比了多次波逆时偏移成像结果与多次波最小二乘逆时偏移成像结果，并对比了多次波最小二乘逆时偏移成像结果与一阶多次波最小二乘逆时偏移成像结果.在这个例子中，多次波和一阶多次波最小二乘逆时偏移都只使用了5次迭代.多次波最小二乘逆时偏移消除了多次波逆时偏移中大部分的串声假象； 而一阶多次波最小二乘逆时偏移提供比多次波最小二乘逆时偏移有更高信噪比的成像结果.虽然一阶多次波最小二乘逆时偏移缺少了高阶多次波的信息，依然能提供与多次波最小二乘逆时偏移相似的有效构造成像结果.图10中, 多次波及一阶多次波最小二乘逆时偏移表现出相似的快速稳定的收敛性质.

4 讨论

利用SRME方法将反射波和多次波分离是针对海上采集数据的常规处理流程之一.SRME方法需要较密的炮检排列和近偏移距数据，因此在使用之前需要做数据规则化；尤其是对于三维数据，横向上的采集数据较为稀疏，增加了数据规则化的难度(Dragoset et al., 2010).另外，在三维数据上使用SRME方法，需要存储大规模的共道集数据.

图8 Marmousi2 声波速度模型Fig.8 Marmousi2 acoustic velocity model

图9 (a)多次波逆时偏移成像结果；(b)多次波最小二乘逆时偏移成像结果(5次迭代)；(c)一阶多次波最小二乘逆时偏移成像结果(5次迭代)

As indicated by white labels in (a) and (b), LSRTM of multiples suppresses most of the crosstalk artifacts in the RTM of multiples. Black labels in (b) and (c) highlight that LSRTM of first-order multiples provides an image with a higher signal to noise ratio than LSRTM of multiples; and they provide similar true images.

图10 Marmousi2模型上多次波(实线)及一阶多次波(散点)最小二乘逆时偏移的归一化数据残差收敛曲线Fig.10 Normalized data residual for LSRTM of multiples (solid line) and first-order multiples (dots) on the Marmousi2 model

常规的数据处理提供了分离的反射波和多次波; 多次波成像利用了传统处理流程中被认为是噪声而丢掉的多次波，提供了除反射波外的额外地下照明.而从所有多次波中分离一阶多次波，无需额外的数据规则化，增加的计算量主要为：通过反射波与多次波的一次褶积来预测除了一阶外的所有高阶多次波，将高阶多次波从所有多次波中减去.一阶多次波的分离方法很容易拓展到三维算法，难点在于增加的存储量和计算量.

图11 在R2位置记录到的反射波偏移成像时，总的传播路径为SX1和R2X1；在R2位置记录到的多次波偏移成像时(Liu et al., 2011a, 2011b)，总的传播路径为R1X2和R2X2Fig.11 When the primary recorded at R2 is migrated, the total propagation path is SX1 and R2X1. When the multiple recorded at R2 is migrated, the total propagation path is R1X2and R2X2

反射波最小二乘逆时偏移需要较好的偏移速度(Dai and Schuster, 2013; Huang et al., 2014).而多次波或者一阶多次波最小二乘逆时偏移，受偏移速度不准确的影响相对较小.因为，在相同的偏移距处，反射波成像比多次波成像的传播路径要更长(如图11).

多次波成像与反射波成像的主要区别在于震源项的不同，而两者的正演算法是相同的.因此多次波或者一阶多次波最小二乘逆时偏移也可以拓展到三维模型上.三维的算法能够使实际资料的偏移归位更加准确，因此，拓展到三维算法能提高多次波或者一阶多次波最小二乘逆时偏移在实际资料应用时的收敛性.

5 结论

多次波逆时偏移成像能够对下地表提供额外的照明，但是却产生了很多串声噪声.我们在数据和成像域验证了最小二乘逆时偏移能够消除多次波逆时偏移产生的串声假象.利用多次波的最小二乘逆时偏移的成像剖面，波恩模拟的多次波与观测的多次波有很好的匹配.然而， 在多次波最小二乘逆时偏移成像剖面中，往往会有残余的噪声.我们利用修改的SRME流程将一阶多次波从所有多次波中分离出后，使用同样的迭代次数，一阶多次波最小二乘逆时偏移能够提供与多次波最小二乘逆时偏移相似的有效构造成像结果；而一阶多次波最小二乘逆时偏移结果中有更少的噪声.总之，多次波或者一阶多次波最小二乘逆时偏移，能够以较高的信噪比为下地表提供额外的照明，或许可以为复杂结构的成像做出贡献.

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(本文编辑 何燕)

Least-squares reverse-time migration of surface-related multiples

LIU Xue-Jian1,2, LIU Yi-Ke1

1KeyLaboratoryofEngineeringGeomechanics,InstituteofGeologyandGeophysics,ChineseAcademyofSciences,Beijing100029,China2UniversityofChineseAcademyofSciences,Beijing100049,China

Surface-related multiples are traditionally treated as noise and are attenuated using surface-related multiples elimination (SRME) and/or radon-based multiple-elimination methods. Multiples penetrate into the subsurface several times and contain abundant reflection information of small angles. Compared with migrating of primaries, migrating of multiples extends all the receivers as second sources and sometimes provides additional subsurface illumination. For reverse-time migration (RTM) of all-order multiples, however, the main challenge is that undesired crosscorrelations between forward and backward propagated seismic waves generate so many crosstalk artifacts. The crosstalks may distribute in the whole image profile, which can destruct the true image of reflectors and mislead the interpreting result of a migrated image. Compared with conventional RTM, least-squares reverse-time migration (LSRTM) can invert recorded primaries as an image with more balanced amplitude and higher resolution. Moreover, we develop the conventional LSRTM to invert multiples as an image while iteratively suppressing crosstalk artifacts. However, LSRTM of multiples can′t totally attenuate the artifacts in the image of multiples, and usually many iterations are required to invert a well-accepted image. Alternatively, if first-order multiples can be separated from all-order multiples in advance, LSRTM of first-order multiples can be developed to reduce the iteration number. With the same iterations used, compared with LSRTM of multiples, LSRTM of first-order multiples can provide a much cleaner image section and a similar true image of reflectors. The motivation to develop LSRTM of first-order multiples can be further summarized as: (1) conventional migration of first-order multiples can avoid the most undesired crosscorrelations between forward and backward propagated wavefields and can maintain some advantages of imaging multiples at the same time; although the subsurface information contributed by higher-order multiples is neglected, RTM of first-order multiples have already avoided most artifacts. (2) There are still some crosstalk artifacts in the RTM image of first-order multiples; then, compared with RTM of first-order multiples, LSRTM of first-order multiples can further enhance the image in detail by suppressing the crosstalk artifacts, balancing the amplitude, and improving the resolution.

Least-Squares Reverse-Time Migration (LSRTM); Migration of multiples; First-order multiples

10.6038/cjg20160919.

国家自然科学基金项目(41430321，41374138)和中国科学院战略性先导科技专项(B类) (XDB01020300)联合资助.

刘学建，男，1987年生，在读博士研究生，从事表面多次波消除、多次波成像方法、最小二乘逆时偏移以及逆时偏移角道集等方面的研究. E-mail:liuxuejian10@mails.ucas.ac.cn

10.6038/cjg20160919

P631

2015-09-01，2016-07-28收修定稿

刘学建，刘伊克. 2016. 表面多次波最小二乘逆时偏移成像. 地球物理学报，59(9):3354-3365,

Liu X J, Liu Y K. 2016. Least-squares reverse-time migration of surface-related multiples.ChineseJ.Geophys. (in Chinese),59(9):3354-3365,doi:10.6038/cjg20160919.